Apple Inc. is preparing to bolster its streaming-music business with a proposed $3.2 billion deal to acquire Beats Electronics LLC, the high-end headphone maker that recently launched the Beats Music subscription service, according to people familiar with the matter.

The potential deal would provide Apple with an entree into two burgeoning markets: high-end headphones and subscription music. Services that charge a monthly fee for access to an unlimited amount of music are growing rapidly, led by Spotify AB, which counts...